In the simplest case, such absorption could be described with a non-Hermitian potential (i.e. one where probability is not conserved).
This is the spectral theorem in mathematics, and in a finite state space it is just a statement of the completeness of the eigenvectors of a Hermitian matrix.
If V is finite dimensional with a given basis, this is equivalent to the condition that the matrix of A is Hermitian, i.e., equal to its conjugate transpose A*.
A density matrix is a Hermitian, positive semi-definite matrix of trace one.
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In 1978 Constantine Callias, at the suggestion of his Ph.D. advisor Roman Jackiw, used the axial anomaly to derive this index theorem on spaces equipped with a Hermitian matrix called the Higgs field.